Problem 28 - Physics for Scientists and Engineers with Modern Physics

Question

Which of the following reactions and decays are possible? For those forbidden, explain what laws are violated. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{n}+\eta^{0}\) (b) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{n}+\pi^{0}\) (c) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+}\) (d) \(\mathrm{p} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}}\) (e) \(\mu^{+} \rightarrow \mathrm{e}^{+}+\bar{\nu}_{\mu}\) \((f) \mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\)

Step-by-Step Solution

Answer

Reactions (a), (e), and (f) are possible; (b), (c), and (d) are forbidden due to violation of conservation laws.

Step 1: Analyze reaction (a)

Consider the reaction \( \pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \). Check the conservation of baryon number, charge, and lepton number. Both initial and final states have a baryon number of 1 and a charge of -1. The reaction \( \eta^{0} \) is considered a neutral meson, thus lepton numbers are zero throughout. This reaction respects all the conservation laws and is possible.

Step 2: Analyze reaction (b)

Consider the reaction \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{n} + \pi^{0} \). The initial baryon number is 1 and the charge is +2, while the final baryon number is also 1 and charge is +1. Baryon number is conserved, but charge is not conserved (+2 is not equal to +1). Therefore, this reaction is forbidden due to charge conservation violation.

Step 3: Analyze reaction (c)

Examine \( \pi^{+} + \mathrm{p} \rightarrow \mathrm{p} + \mathrm{e}^{+} \). Initially, the baryon number is 1 and charge is +2; finally, the baryon number is 1 and charge is +2. However, the lepton number before is 0 (no leptons present) and becomes +1 on the right (due to the positron). Therefore, lepton number is not conserved, making this reaction impossible.

Step 4: Analyze decay (d)

Consider \( \mathrm{p} \rightarrow \mathrm{e}^{+} + u_{\mathrm{e}} \). The proton has a baryon number of 1. After decay, there are no baryons, violating baryon number conservation. Additionally, initial lepton number is 0, while the final is \(+1 - 1 = 0\). Despite lepton conservation, the baryon number violation makes this decay impossible.

Step 5: Analyze decay (e)

Evaluate \( \mu^{+} \rightarrow \mathrm{e}^{+} + \bar{u}_{\mu} \). The initial lepton number is +1 (due to the muon), and the final number remains +1 (due to the electron and anti-muon neutrino). All conservation laws are respected, making this decay possible.

Step 6: Analyze decay (f)

Consider \( \mathrm{p} \rightarrow \mathrm{n} + \mathrm{e}^{+} + u_{\mathrm{e}} \). The initial baryon number is 1 and charge is +1. After decay, the baryon number is still 1 (due to the neutron), charge is +1, and the lepton number is zero. All conservation laws hold, making this decay possible.

Key Concepts

Baryon Number Conservation

In particle physics, keeping track of the baryon number is crucial. Baryons, like protons and neutrons, are particles composed of three quarks. Baryon number conservation means the total baryon count remains the same in reactions and decays. Each baryon contributes a number of +1 to this count, while antibaryons contribute -1. This way, it ensures that no baryons mysteriously appear or disappear during a reaction.

For example, when examining the reaction \(\pi^{-} + \mathrm{p} \rightarrow \mathrm{n} + \eta^{0} \), both sides of the reaction have the same baryon number. This indicates the reaction is possible. But, for \(\mathrm{p} \rightarrow \mathrm{e}^{+} + u_{\mathrm{e}}\), the lost baryon number on the right side suggests a forbidden decay. In this case, the baryon number law tells us this decay violates the rule.

• Each baryon adds +1, antibaryon -1.
• Reactions must conserve the total baryon number.
• Discrepancy indicates violation of baryon number conservation.

Charge Conservation

Charge conservation is another essential law in particle physics. It ensures that the total electric charge remains constant in any physical process. Every particle comes with a specific charge, which can be positive, negative, or neutral.

Consider the encounter \(\pi^{+} + \mathrm{p} \rightarrow \mathrm{n} + \pi^{0} \). Here, the start shows a total charge of +2, but the end shows only +1. This mismatch reveals a violation of charge conservation, rendering the reaction impossible. Charge conservation plays a critical role in identifying which particle transformations can naturally occur.

• Charge must be the same before and after the reaction.
• A mismatch flags an impossible reaction.
• Used to check the validity of many particle interactions.

Lepton Number Conservation

In the realm of particle physics, lepton number conservation is equally vital. Leptons, like electrons and neutrinos, each carry a lepton number. Scientists use this concept to ensure particles behave predictably in reactions.

Take the scenario \(\pi^{+} + \mathrm{p} \rightarrow \mathrm{p} + \mathrm{e}^{+} \). Initially, the lepton number is zero, but after the reaction, a positron increases it to +1, hinting at a lepton number violation. Hence, this law concludes that the reaction cannot naturally occur. Lepton number conservation acts as a checkpoint to validate reactions, maintaining the balance between leptons and antileptons.

• Leptons contribute +1, antileptons -1.
• Lepton number must remain unchanged.
• Violations indicate a forbidden occurrence.